The $K$-sign depth ($K$-depth) of a model parameter $\theta$ in a data set is the relative number of $K$-tuples among its residual vector that have alternating signs. The $K$-depth test based on $K$-depth, recently proposed by Leckey et al. (2020), is equivalent to the classical residual-based sign test for $K=2$, but is much more powerful for $K\ge 3$. This test has two major drawbacks. First, the computation of the $K$-depth is fairly time consuming having a polynomial time complexity of degree $K$, and second, the test requires knowledge about the quantiles of the test statistic which previously had to be obtained by simulation for each sample size individually. We tackle both of these drawbacks by presenting a limit theorem for the distribution of the test statistic and deriving an (asymptotically equivalent) form of the $K$-depth which can be computed efficiently. For $K=3$, such a limit theorem was already derived in Kustosz et al. (2016a) by mimicking the proof for $U$-statistics. We provide here a much shorter proof based on Donsker’s theorem and extend it to any $K\ge 3$. As part of the proof, we derive an asymptotically equivalent form of the K-depth which can be computed in linear time. This alternative and the original implementation of the K-depth are compared with respect to their runtimes and absolute difference.

这 $ķ$信号深度（$ķ$深度） $\theta$ 数据集中的相对数量 $ķ$其残差向量中具有交替符号的元组。这$ķ$深度测试 $ķ$深度，最近由Leckey等人提出。（2020），等效于经典的基于残差的符号测试$ķ=\mathrm{2个}$，但功能强大得多 $ķ\ge 3$。该测试有两个主要缺点。首先，计算$ķ$深度是相当耗时的，具有度数的多项式时间复杂度 $ķ$其次，测试需要有关测试统计量的分位数的知识，以前必须通过模拟分别为每个样本大小来获得分位数。我们通过提出检验统计量分布的一个极限定理并推导该检验统计量的（渐近等价）形式，解决了这两个缺点。$ķ$深度，可以有效地进行计算。为了$ķ=3$，这样的极限定理已经在Kustosz等人中得到了。（2016a）通过模仿$ü$-统计数据。我们在此根据Donsker定理提供一个简短得多的证明，并将其扩展到任何$ķ\ge 3$。作为证明的一部分，我们得出了K深度的渐近等效形式，可以在线性时间内计算出该形式。比较了这种替代方法和K深度的原始实现，并比较了它们的运行时间和绝对差异。